Group theory and generalizations Definition
Normal subgroup
Builds on SubgroupConjugate subgroup
Definition
Let be a group and let be a subgroup of . The subgroup is normal in when for all and all . The notation is .
Remarks
Equivalently, for each , that is, equals each of its conjugates. Each group has the normal subgroups and . In the symmetric group , the subgroup is normal, but the subgroup is not.
Reading path
To understand this page from the ground up, read these 5 pages in order. The foundations come first.
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- Group concept
- Conjugate elements definition
- Subgroup definition
- Conjugate subgroup definition
- Normal subgroup definition
Where this leads
100 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Semidirect product
- Normality via left and right cosets
- Extensions of solvable groups are solvable
- Quotients of solvable groups are solvable
- Inner automorphism group is isomorphic to G modulo its centre
- Subgroups of solvable groups are solvable
- Classification of finite simple groups
- Finite p-groups are nilpotent
- Class equation
- Subnormal series
- Fixed points of a p-group action
- Second Sylow theorem
- Outer automorphism group
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- If G modulo its centre is cyclic then G is abelian
- Simplicity of the alternating group
- Action on left cosets
- Simple group
- The kernel is a normal subgroup
- First isomorphism theorem
- Upper central series
- Nilpotent group
- Jordan–Hölder theorem
- Orbit-stabiliser theorem
- Universal property of the abelianisation
- A subgroup whose index is the smallest prime divisor of the order is normal
- Burnside's p^a q^b theorem
- Universal property of the quotient group
- The alternating group A5 is simple
- Nilpotent groups are solvable
- Abelianisation
- The symmetric group on five or more letters is not solvable
- Abelian simple groups have prime order
- Cauchy's theorem
- Subgroups of abelian groups are normal
- The number of Sylow subgroups divides the index
- A quotient is abelian iff the subgroup contains the commutator subgroup
- Composition series
- Solvable group
- Second isomorphism theorem
- Internal direct product
- Third Sylow theorem
- Von Dyck's theorem
- Frattini argument
- Presentation of a group
- Abelian groups are solvable
- Composition factors
- Torsion subgroup
- Structure theorem for finitely generated abelian groups
- A nontrivial finite p-group has nontrivial centre
- Size of an orbit times the size of the stabiliser
- Correspondence theorem
- Characteristic subgroups are normal
- Order of the alternating group
- Normal core
- Third isomorphism theorem
- Product of a subgroup and a normal subgroup is a subgroup
- Subgroups of index two are normal
- First Sylow theorem
- Size of a conjugacy class is the index of the centraliser
- Groups of order pq
- Integers modulo n as a quotient group
- Free abelian group
- Quotient group
- Burnside's lemma
- Abel–Ruffini theorem
- Zassenhaus lemma
- Galois group of a prime degree polynomial with two nonreal roots
- Proof that subgroups of index two are normal
- The integers modulo the even integers
- The even integers form a subgroup of the integers
- Hall's theorem on solvable groups
- Krull–Schmidt theorem
- Rational equivalence
- Proof that the kernel is a normal subgroup
- Intersection multiplicity
- Hall subgroup
- Fitting's lemma
- Sylow theorems
- Frattini subgroup
- Lattice-ordered group
- Proof of the first isomorphism theorem
- Jordan–Hölder theorem for modules
- The cosets of a subgroup that is not normal do not form a group
- Algebraic cycle
- Algebraic K-theory of schemes
- Structure theorem for compactly generated LCA groups
- Left-orderable group
- Schreier refinement theorem
- Solvability of x^n - a
- Quintic equation
- Shafarevich's theorem on solvable Galois groups
- Length of a module
- Galois' theorem on solvability
- Torsion-free group
- Divisor class group
- Solvable by radicals
- Orderable group
Further reading
- Web https://en.wikipedia.org/wiki/Normal_subgroup, Definitions
About this page
| Tag | 0W77CVBCHQ |
|---|---|
| URI | m2://0W77CVBCHQ |
| Kind | definition |
| Writing | W2 usable |
| MSC | 20A05 |
| Defines | Normal subgroup |
| Uses | Subgroup, Conjugate subgroup |
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What links here
Used by
- Semidirect product
- Normality via left and right cosets
- Subnormal series
- Simplicity of the alternating group
- Simple group
- The kernel is a normal subgroup
- Nilpotent group
- A subgroup whose index is the smallest prime divisor of the order is normal
- Subgroups of abelian groups are normal
- A quotient is abelian iff the subgroup contains the commutator subgroup
- Internal direct product
- Presentation of a group
- Correspondence theorem
- Characteristic subgroups are normal
- Normal core
- Product of a subgroup and a normal subgroup is a subgroup
- Quotient group
- The even integers form a subgroup of the integers
- Proof that the kernel is a normal subgroup
- The cosets of a subgroup that is not normal do not form a group